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Explore recent mathematical research on discreteness and freeness properties of 2-generator subgroups within the special linear group SL_2(K) across different local fields in this 11-minute conference talk. Discover findings from collaborative work examining the non-archimedean case with Matthew Conder, real number applications through student research by Ari Markowitz, and complex number investigations conducted with Alex Elzenaar and Gaven Martin. Gain insights into advanced group theory and algebraic structures as the speaker presents a comprehensive overview of these specialized mathematical results across various field contexts.